The tree augmentation problem (TAP) is a fundamental network design problem,
in which the input is a graph G and a spanning tree T for it, and the goal
is to augment T with a minimum set of edges Aug from G, such that T∪Aug is 2-edge-connected.
TAP has been widely studied in the sequential setting. The best known
approximation ratio of 2 for the weighted case dates back to the work of
Frederickson and J\'{a}J\'{a}, SICOMP 1981. Recently, a 3/2-approximation was
given for unweighted TAP by Kortsarz and Nutov, TALG 2016. Recent breakthroughs
give an approximation of 1.458 for unweighted TAP [Grandoni et al., STOC 2018],
and approximations better than 2 for bounded weights [Adjiashvili, SODA 2017;
Fiorini et al., SODA 2018].
In this paper, we provide the first fast distributed approximations for TAP.
We present a distributed 2-approximation for weighted TAP which completes in
O(h) rounds, where h is the height of T. When h is large, we show a
much faster 4-approximation algorithm for the unweighted case, completing in
O(D+nlog∗n) rounds, where n is the number of vertices and D is
the diameter of G.
Immediate consequences of our results are an O(D)-round 2-approximation
algorithm for the minimum size 2-edge-connected spanning subgraph, which
significantly improves upon the running time of previous approximation
algorithms, and an O(hMST+nlog∗n)-round 3-approximation
algorithm for the weighted case, where hMST is the height of the MST of
the graph. Additional applications are algorithms for verifying
2-edge-connectivity and for augmenting the connectivity of any connected
spanning subgraph to 2.
Finally, we complement our study with proving lower bounds for distributed
approximations of TAP